What is a Competitive Equilibrium?

These notes cover chapters 4 and 5 from Williamson.

In these chapters, we build a simple economic model called a competitive equilibrium. Note that "competitive equilibrium" doesn't refer to a single specific model, but rather refers to a class of economic models, which all share some commonalities. If we write one of these models out, it will look something like this:

A Vague Competitive Equilibrium:

Given {some set of exogenous parameters}, our {set of endogenous parameters} will be in competitive equilibrium if the following conditions are satisfied:

Let's go into a bit more detail about each of these pieces. The key features of a competitive equilibrium model are as follows:

Making assumptions about the world

To build a model like this, we have to make assumptions about which agents are in the economy, what markets they are interacting with, what kinds of choices each agent can make, etc. The details of these assumptions will change the structure of our model. Then if the model behaves in a way which is consistent with the real world, it hints that our assumptions may be reasonable.

But more importantly, if the model makes bad predictions, and doesn't match up with what we see, this suggests that one or more of our assumptions are innacurate. And this can be just as informative, if not more so.

The point of representing the economy algebraically in models like these isn't that math-notation makes an argument true. But rather that math-notation forces us to be rigourous and specific about our assumptions. And this allows us to more meaningfully evaluate and learn from those assumptions.

In chapters 4 and 5, we build a very simple competitive equilibrium. And we make many assumptions to simplify the math. The point of these chapters isn't to have you memorize the specific assumptions we are using in this model and treat them as gospel truth about the world. Rather, the aim is to practice the process of taking a description of the economy and expressing it as a model like these ones.

In later chapters, we will change our assumptions about the structure of the economy, and build different competitive equilibrium models. For example, we might open the economy to trade, or add multiple time periods.

So what is our model for these chapters?

In the model for chapters 4 and 5, we make plenty of assumptions about the economy that definitely aren't true. For example:

Both of these first two assumptions are fairly reasonable for looking at aggregate economic activity. We will add multiple time periods into the model in the second half of the course. And if we have time near the end of the semester, we may look at what happens when we change our assumptions about the role of prices.

The model for chapters 4 and 5, in it's full form, looks like this:

A Closed-Economy One-Period Macroeconomic Model, with Lump-sum Taxes:

Given exogenous \(\left\{ h,z,K,g \right\}\), a competitive equilibrium is a set of allocations \(\left\{ C,l,N_s,N_d,T \right\}\) and a real wage \(\left\{ w \right\}\) which satisfy the following:

In the above model, we are using the following variables to represent different things in our economy:

And we can write out the model like so:

Let's go a bit more in detail about each of these pieces.

The Agents:

In this class, we look at several simple examples of a competitive equilibrium.

	Market Clearing

Note that on a test, the most important thing is that you have each of the conditions: Maximization for each agent, government budget, and market clearing. The preface is needed for strict rigour, but I won't be too harsh about it.